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Triangle T(n, k) = n^(n-1) * Fibonacci(k)^(n+1) - (n-1)! * (Fibonacci(k) - 1) * Sum_{j=0..n} (n*Fibonacci(k))^j/j!, with T(n, 0) = n! and T(n, 1) = n^(n-1), read by rows.
3

%I #8 Jan 06 2022 02:30:32

%S 1,1,1,2,2,2,6,9,9,22,24,64,64,266,708,120,625,625,4536,17457,108129,

%T 720,7776,7776,100392,563088,5709120,52517688,5040,117649,117649,

%U 2739472,22516209,375217945,5489293264,92757410569,40320,2097152,2097152,89020752,1076444064,29566405440,688833593904,18867973329344,513683908057152

%N Triangle T(n, k) = n^(n-1) * Fibonacci(k)^(n+1) - (n-1)! * (Fibonacci(k) - 1) * Sum_{j=0..n} (n*Fibonacci(k))^j/j!, with T(n, 0) = n! and T(n, 1) = n^(n-1), read by rows.

%H G. C. Greubel, <a href="/A137227/b137227.txt">Rows n = 0..50 of the triangle, flattened</a>

%F T(n, k) = (1/n)*( n^n * Fibonacci(k)^(n+1) - n! * (Fibonacci(k) - 1) * Sum_{j=0..n} (n*Fibonacci(k))^j/j! ), with T(n, 0) = n! and T(n, 1) = n^(n-1).

%e Triangle begins as:

%e 1;

%e 1, 1;

%e 2, 2, 2;

%e 6, 9, 9, 22;

%e 24, 64, 64, 266, 708;

%e 120, 625, 625, 4536, 17457, 108129;

%e 720, 7776, 7776, 100392, 563088, 5709120, 52517688;

%e 5040, 117649, 117649, 2739472, 22516209, 375217945, 5489293264, 92757410569;

%t T[n_, k_]:= If[k==0, n!, If[k==1, n^(n-1), (1/n)*(Fibonacci[k]^(n+1)*n^n - n!*(Fibonacci[k] -1)*Sum[n^j*Fibonacci[k]^j/j!, {j,0,n}])]];

%t Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jan 06 2022 *)

%o (Sage)

%o @CachedFunction

%o def A137227(n,k):

%o if (k==0): return factorial(n)

%o elif (k==1): return n^(n-1)

%o else: return (1/n)*(fibonacci(k)^(n+1)*n^n - factorial(n)*(fibonacci(k) -1)*sum((n*fibonacci(k))^j/factorial(j) for j in (0..n)))

%o flatten([[A137227(n,k) for k in (0..n)] for n in (0..10)]) # _G. C. Greubel_, Jan 06 2022

%Y Cf. A000045, A122525, A137216.

%K nonn,tabl

%O 0,4

%A _Roger L. Bagula_, Mar 07 2008

%E Edited by _G. C. Greubel_, Jan 06 2022